Projective Planes of Order 49 Related to t63

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t63 and all known planes of order 49 obtained from it by dualizing and deriving. Coming soon: also planes related by the method of lifting quotients. This list is currently incomplete; check back later for a complete enumeration.

Following the table is a key to the table.

Known Projective Planes of Order 49 Related to t63

Entry Plane |Autgp| Point Orbits Line Orbits 7-rank
1 Translation Plane t63, dual dt63 1382976 2,8,122,16,2401 1,98,392,5882,784 927
2 t63_0_0, dt63_0_0 6174 1,78,2116,2058 12,32,42,497,14714 985
3 t63_0_1, dt63_0_1 2058 1,756,2058 18,42,4949 985
4 t63_1_0, dt63_1_0 2058 1,756,2058 18,42,4949 985
5 t63_2_0, dt63_2_0 4116 1,78,1424,2058 12,23,42,497,9821 985
6 t63_2_1, dt63_2_1 4116 1,78,1424,2058 12,23,42,497,9821 985
7 t63_3_0, dt63_3_0 4116 1,78,1424,2058 12,23,42,497,9821 985
8 t63_3_1, dt63_3_1 12348 1,72,143,212,427,2058 12,6,42,49,983,1472,2946 983
9 t63_4_0, dt63_4_0 12348 1,72,143,212,427,2058 12,32,42,49,983,2947 979

Key to the table

Only one line is displayed for both a plane and its dual, an asterisk (*) in the first column indicating that the plane is self-dual. Each line includes the following information and isomorphism invariants for each plane.

• Plane    provides a pzip file containing the projective plane. I assume you have installed pzip under linux, which compresses each plane to about 6 KB. Right-click to save as plane.pz, then type the command punzip plane.pz to recover plane as a text file. This text file has 2451 rows, each row specifying a line of the plane as a subset of the points 0,1,2,...,2450. For non-self-dual planes, a second file is also given, containing the dual of the first.
• |Autgp|    The order of the full collineation group of the projective plane. This table entry is linked to a gzip file providing generators of the automorphism group. Right-click to save as plane.gz, then type the command gunzip plane.gz to recover plane as a text file. This text file lists generators of the full collineation group as permutations of the integers 0,1,2,...,4901 where 0,1,2,...,2450 are labels for the points and 2451,...,4901 are labels for the lines. Generators for the automorphism group of the dual plane are not provided since these are trivially obtained from the generators given for the original plane.
• Point Orbits    The lengths of the full automorphism group on the points. (These are the lengths of the line orbits for the dual plane.)
• Line Orbits    The lengths of the full automorphism group on the lines. (These are the lengths of the point orbits for the dual plane.)
• 7-rank    The rank of the (0,1)-incidence matrix of the projective plane, over a field of characteristic 7.

/ revised February, 2011